Shapes, Fractals, Time & the Dimensions they Belong to

First, second and third dimensions, and why fractals don’t belong to any of them, as well as what happens when you get into higher dimensions.

Foreword

In this post, I’m going to try my best to explain the first, second, and third dimensions, and why fractals don’t belong to any of them, as well as what happens when you get into higher dimensions. But before getting into the nitty-gritty of the subject, I think it’s worth prefacing this post with a short note on the nature of mathematics itself:

Alain Badiou said that mathematics is a rigorous aesthetic; it tells us nothing of real being, but forges a fiction of intelligible consistency. That being said, I think it’s interesting to think about whether or not mathematics were invented or discovered – whether or not numbers exist outside of the human mind.

While I don’t have an answer to this question (and there are at least three different schools of thought on the subject), I do think it’s important to keep in mind that we only use math as a tool to measure and represent ‘real world’ things. In other words, our knowledge of mathematics has its limitations as far as understanding the space-time continuum goes.

 

1.0 Traditional Dimensions

In physics and mathematics, dimensions are used to define the Cartesian plains. The measure of a mathematical space is based on the number of variables require to define it. The dimension of an object is defined by how many coordinates are required to specify a point on it.

It’s important to note that there is no ‘first’ or ‘second’ dimension. It’s a bit like pouring three cups of water into a vase and asking someone which cup is the first one. The question doesn’t even make sense.

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Excerpt from ‘minutephysics’
We usually arbitrarily pick a dimension and calling it the ‘first’ one.

 

1.1 – Zero Dimensions

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Something of zero dimensions give us a point. While a point can inhabit (and be defined in) higher dimensions, the point itself has a dimension of zero; you cannot move anywhere on a point.

1.2 – One Dimension

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A line or a curve gives us a one-dimensional object, and is typically bound by two zero-dimensional things.
Only one coordinate is required to define a point on the curve.
Similarly to the point, a curve can inhabit higher dimension (i.e. you can plot a curve in three dimensions), but as an object, it only possesses one dimension.
Another way to think about it is: if you were to walk along this curve, you could only go forwards or backward – you’d only have access to one dimension, even though you’d be technically moving through three dimensions.

 

1.3 – Two Dimensions

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Surfaces or plains gives us two-dimensional shapes, and are typically bound by one-dimensional shapes (lines/curves).

A plain can be defined by x&yy&z or x&z; more complex surfaces are commonly defined by u&v values. These variable are arbitrary, what is important is that there are two of them.

A surface can live in three+ dimensions, but still only possesses two dimension. Two coordinate are required to define a point on a surface. For example a sphere is a three-dimensional object, but the surface of a sphere is two-dimensional – a point can be defined on the surface of a sphere with latitude and longitude.

 

1.4 – Three Dimensions

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A volume gives us a three-dimensional shape, and can be bound by two-dimensional shapes (surfaces).

Shapes in three dimensions are most commonly represented in relation to an x, y and z axis. If a person were to swim in a body of water, their position could be defined by no less than three coordinates – their latitude, longitude and depth. Traveling through this body of water grants access to three dimensions.

 

2.0 Fractal Dimensions

Fractals can be generally classified as shapes with a non-integer dimension (a dimension that is not a whole number). They may or may not be self-similar, but are typically measured by their properties at different scales.

Felix Hausdorff and Abram Besicovitch demonstrated that, though a line has a dimension of one and a square a dimension of two, many curves fit in-between dimensions due to the varying amounts of information they contain. These dimensions between whole numbers are known as Hausdorff-Besicovitch dimensions.

 

2.1 – Between the First & Second Dimensions

A line or a curve gives us a one-dimensional object that allows us to move forwards and backwards, where only one coordinate is required to define a point on them.

Surfaces give us two-dimensional shapes, where two coordinate are required to define a point on them.

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Here is a shape that cannot be classified as a one-dimensional shape, or a two-dimensional shape. It can be plotted in two dimensions, or even three dimensions, but the object itself does not have access the two whole dimensions.

If you were to walk along the shape starting from the base, you could go forwards and backwards, but suddenly you have an option that’s more than forwards and backwards, but less than left and right.

You cannot define a point on this shape with a single coordinate, and a two coordinate system would define a point off of the shape more often than not.

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Each fractal has a unique dimensional measure based on how much space they fill.

 

2.2 – Between the Second & Third Dimensions

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Developing Koch Snowflake
The same logic applies when exploring fractals plotted in three dimensions:

Surfaces give us two-dimensional shapes, where two coordinate are required to define a point on them.

A volume gives us a three-dimensional shape where a point could be defined by no less than three coordinates.

While these models live in three dimensions, they do not quite have access to all of them. You cannot define a point on them with two coordinates: they are more than a surface and less than a volume.

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Fourth Iteration Menger Sponge
The Menger Sponge for example has (mathematically) a volume of zero, but an infinite surface area.

 

2.3 – Calculating Fractal Dimensions

The following are three methods of calculating Hausdorff-Besicovitch dimension:
• The exactly self-similar method for calculating dimensions of mathematically generated repeating patterns.
• The Richardson method for calculating a dimensional slope.
• The box-counting method for determining the ratios of a fractal’s area or volume.

 

On another note:

In theory, higher (non-integer) dimensional fractals are possible.
As far as I’m concerned however, they’re not particularly good for anything in a three-dimensional world. You are more than welcome to prove me wrong though.

 

3.0 Higher Dimensions

Sadly, living in a three-dimensional world makes it especially difficult to think about, and nearly impossible to visualise, higher dimensions. This is in the same way that a two-dimensional being would find it impossibly hard to think about our three-dimensional world, a subject explored in the novel ‘Flatland’ by Edwin A. Abbott.

That being said, it’s plausible that we experience much higher dimensions that are just too hard to perceive. For example, an ant walking along the surface of a sphere will only ever perceive two dimensions, but is moving through three dimensions, and is subject to the fourth (temporal) dimension.

 

3.1 – The Fourth Dimension (Temporal)

If we consider time an additional variable, then despite the fact that we live in a three dimensional world, we are always subject to (even if we cannot visualize) a fourth dimension.

Neil deGrasse Tyson puts it quite plainly by saying:
“[…] you have never met someone at a place, unless it was at a time; you have never met someone at a time, unless it was at a place […]”

Suppose we call our first three dimensions x, y & z, and our fourth t: latitude, longitude, altitude and time, respectively. In this instance, time is linear, and time & space are one. As if the universe is a kind of film, where going forwards and backwards in time will always yield the exact same outcome; no matter how many times you return to a point in point time, you will always find yourself (and everything else) in the exact same place.
However time is only linear for us as three-dimensional beings. For a four-dimensional being, time is something that can be moved through as freely as swimming or walking.
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Excerpt from ‘Seeker’

 

3.2 – The Fourth Dimension (Spacial)

If we explore spacial dimensions, a four-dimensional object may be achieved by ‘folding’ three-dimensional objects together. They cannot exist in our three-dimensional world, but there are tricks to visualise them.

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We know that we can construct a cube by folding a series of two-dimensional surfaces together, but this is only possible with the third dimension, which we have access to.

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If we visualise, in two dimensions, a cube rotating (as seen above), it looks like each surface is distorting, growing and shrinking, and is passing through the other. However we are familiar enough with the cube as a shape to know that this is simply a trick of perspective – that objects only look smaller when they are farther away.

In the same way that a cube is made of six squares, a four-dimensional cube (hypercube or tesseract), is made of eight cubes.

  • A line is bound by two zero-dimensional things
  • A square is bound by four one-dimensional shapes
  • A cube is bound by six two-dimensional surfaces
  • A hypercube, bound by eight three-dimensional volumes

It looks like each cube is distorting, growing and shrinking, and passing through the other. This is because we can only represent eight cubes folding together in the fourth dimension with three-dimensional perspective animation.

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3D representation of eight cubes folding in 4D space to form a Hypercube

Perspective makes it look like the cubes are growing and shrinking, when they are simply getting closer and further in four-dimensional space. If somehow we could access this higher dimension, we would see these cubes fold together unharmed the same way forming a cube leaves each square unharmed.

Below is a three-dimensional perspective view of hypercube rotating in four dimensions, where (in four-dimensional space) all eight cubes are always the same, but are being subjected to perspective.

 

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3.3 – The Fifth and Sixth Dimensions

On the temporal side of things, adding the ability to move ‘left & right’ and ‘up & down’ in time gives us the fifth and sixth dimensions.

(For example: x, y, z, t1, t2, t3)

This is a space where one can move through time based on probability and permutations of what could have been, is, was, or will be on alternate timelines. For any one point in this space, there are six coordinates that describe its position.

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In spacial dimensions, it is theoretically possible to fold four-dimensional objects with a fifth dimension. However, it becomes increasingly difficult for us to visualise what is happening to the shapes that we’re folding.
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In theory, objects can keep being folded together into higher and higher spacial dimensions indefinitely. (R1, R2, R3,R4,R5, R6, Retc.)

There’s a terrific explanation of what happens to platonic solids and regular polytopes in higher dimensions on Numberphile: https://youtu.be/2s4TqVAbfz4

 

3.4 – Even Higher Dimensions

If we can take a point and move it through space and time, including all the futures and pasts possible, for that point, we can then move along a number line where the laws of gravity are different, the speed of light has changed.

Dimensions seven though ten are different universes with different possibilities, and impossibilities, and even different laws of physics. These grasp all the possibilities and permutations of how each universe operates, and the whole of reality with all the permutations they’re in, throughout all of time and space. The highest dimension is the encompassment of all of those universes, possibilities, choices, times, places all into a single ‘thing.’

These ten time-space dimensions belong to something called Super-string Theory, which is what physicists are using to help us understand how the universe works.

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Cross section of the quintic Calabi–Yau manifold
There may very well be a link between temporal dimensions and spacial dimensions. For all I know, they are actually the same thing, but thinking about it for too long makes my head hurt. If the topic interests you, there is a philosophical approach to the nature of time called ‘eternalism’, where one may find answers to these questions. Other dimensional models include M-Theory, which suggests there are eleven dimensions.

While we don’t have experimental or observational evidence to confirm whether or not any of these additional dimensions really exist, theoretical physicists continue to use these studies to help us learn more about how the universe works. Like how gravity affects time, or the higher dimensions affect quantum theory.

The Curves of Life

“An organism is so complex a thing, and growth so complex a phenomenon, that for growth to be so uniform and constant in all the parts as to keep the whole shape unchanged would indeed be an unlikely and an unusual circumstance. Rates vary, proportions change, and the whole configuration alters accordingly.” – D’Arcy Wentworth Thompson

“This is the classic reference on how the golden ratio applies to spirals and helices in nature.” – Martin Gardner

The Curves Of Life

What makes this book particularly enjoyable to flip through is an abundance of beautiful hand drawings and diagrams. Sir Theodore Andrea Cook explores, in great detail, the nature of spirals in the structure of plants, animals, physiology, the periodic table, galaxies etc. – from tusks, to rare seashells, to exquisite architecture.

He writes, “a staircase whose form and construction so vividly recalled a natural growth would, it appeared to me, be more probably the work of a man to whom biology and architecture were equally familiar than that of a builder of less wide attainments. It would, in fact, be likely that the design had come from some great artist and architect who had studied Nature for the sake of his art, and had deeply investigated the secrets of the one in order to employ them as the principles of the other.

Cook especially believes in a hands-on approach, as oppose to mathematic nation or scientific nomenclature – seeing and drawing curves is far more revealing than formulas.

252264because I believe very strongly that if a man can make a thing and see what he has made, he will understand it much better than if he read a score of books about it or studied a hundred diagrams and formulae. And I have pursued this method here, in defiance of all modern mathematical technicalities, because my main object is not mathematics, but the growth of natural objects and the beauty (either in Nature or in art) which is inherent in vitality.

Despite this, it is clear that Theodore Cook has a deep love of mathematics. He describes it at the beautifully precise instrument that allows humans to satisfy their need to catalog, label and define the innumerable facts of life. This ultimately leads him into profoundly fascinating investigations into the geometry of the natural world.

 

Relevant Material

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“An organism is so complex a thing, and growth so complex a phenomenon, that for growth to be so uniform and constant in all the parts as to keep the whole shape unchanged would indeed be an unlikely and an unusual circumstance. Rates vary, proportions change, and the whole configuration alters accordingly.” – D’Arcy Wentworth Thompson

D’Arcy Wentworth Thompson wrote, on an extensive level, why living things and physical phenomena take the form that they do. By analysing mathematical and physical aspects of biological processes, he expresses correlations between biological forms and mechanical phenomena.

He puts emphasis on the roles of physical laws and mechanics as the fundamental determinants of form and structure of living organisms. D’Arcy describes how certain patterns of growth conform to the golden ratio, the Fibonacci sequence, as well as mathematics principles described by Vitruvius, Da Vinci, Dürer, Plato, Pythagoras, Archimedes, and more.

While his work does not reject natural selection, it holds ‘survival of the fittest’ as secondary to the origin of biological form. The shape of any structure is, to a large degree, imposed by what materials are used, and how. A simple analogy would be looking at it in terms of architects and engineers. They cannot create any shape building they want, they are confined by physical limits of the properties of the materials they use. The same is true to any living organism; the limits of what is possible are set by the laws of physics, and there can be no exception.

 

Further Reading:

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Biomimicry in Architecture by Michael Pawlyn

“You could look at nature as being like a catalogue of products, and all of those have benefited from a 3.8 billion year research and development period. And given that level of investment, it makes sense to use it.” – Michael Pawlyn

Michael Pawlyn, one of the leading advocates of biomimicry, describes nature as being a kind of source-book that will help facilitate our transition from the industrial age to the ecological age of mankind. He distinguishes three major aspects of the built environment that benefit from studying biological organisms:

The first being the quantity on resources that use, the second being the type of energy we consume and the third being how effectively we are using the energy that we are consuming.

Exemplary use of materials could often be seen in plants, as they use a minimal amount of material to create relatively large structures with high surface to material ratios. As observed by Julian Vincent, a professor in Biomimetics, “materials are expensive and shape is cheap” as opposed to technology where the inverse is often true.

Plants, and other organisms, are well know to use double curves, ribs, folding, vaulting, inflation, as well as a plethora of other techniques to create forms that demonstrate incredible efficiency.

Adaptable Hypars

 

An exploration of the simplest Hyperbolic Paraboloidic ‘saddle’ form has lead to the development of a modular system that combines the principles of the hypar (Hyperbolic Paraboloid) and elastic potential energy.

A hyperbolic paraboloid is an infinite doubly ruled surface in three dimensions with hyperbolic and parabolic cross-sections. It can be parametrized using the following equations:

Mathematical:   z = x2 – yor  x = y z

Parametric:   x(u,v)=u   y(u,v)=v   z(u,v)=uv

The physical manifestation of the above equations can be achieved by constructing a square and forcing the surface area to minimalise by introducing cross bracing that has shorter lengths than the  square edges.

Hypar01Hypar02

A particular square hypar defined by b = n * √2 (b=boundary, n=initial geometry or ‘cross bracing’) thus constricting the four points to the corners of a cube leads to interesting tessellations in three dimensions.

Hypar03Hypar04

Using a simple elastic lashing system to construct a hypar module binds all intersections together whilst allowing rotational movement. The rotational movement at any given intersection is proportionally distributed to all others. This combined with the elasticity of the joints means that the module has elastic potential energy (spring-like properties) therefore an array of many modules can adopt the same elastic properties.

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The system can be scaled, shaped, locked and adapted to suit programmatic requirements.

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Mesh Recursive Subdivision based on Alain Fournier’s algorithm

The principle of the subdivision method to generate mountains is to recursively subdivide (split) polygons of a model up to a required level of detail. At the same time the parts of the split polygons will be perturbed. The initial shape of the model is retained to an extent, depending on the perturbations. Thus, a central point of the fractal subdivision algorithm is perturbation as a function of the subdivision level.

Concerning mountains, the higher the level the smaller the perturbation, otherwise the mountains would get higher and higher. In addition there must be a random number generator to obtain irregularities within the shape – and to achieve a kind of statistical similarity:

P_n = p( n ) * rnd();

Where p_n = perturbation at level n,
P( ) = perturbation function depending on level n, and
Rnd( ) = random number generator.

Fournier developed a subdivision algorithm for a triangle. Here, the midpoints of each side of the triangle are connected, creating four new sub triangles.

Based on this algorithm, the process is done recursively to all the new triangles generated so that the shape is not limited to vertical mountains.

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Random perturbation is where the first iteration is based on a random parameter within the range of 0-9 and the following iterations also are based on a random parameter. This is done using grasshopper by setting the seed number of the initial polygon and the seed number of the iterations. All iterations perturbed based on the z-axis of the new polygon produced.

This resulted in a different shape of the ‘base’ and all the iterations after the first one, ranging from small to big volume depending on the seed of the random number generated. By producing polygons using random perturbation, each iteration is different than the others. The iteration runs for ten (10) times using grasshopper.

Base perturbation * random seed: 1 – 10

Second – third perturbation * random seed: 0 – 10

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Based on the principle, one module is chose to continue with the next step. The module chosen is the 5;5 module which is Base perturbation * random seed: 5 and Second – third perturbation * random seed: 5. After the second iteration, whenever there are surfaces which will intersect, one or both of the surface is removed. This resulted in a random yet in control structure based on the principle applied.

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EXTRA

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For the next brief, I am developing the module to grow further than the third iteration and grows following a certain flow.